Integralen av $$$\sqrt{\frac{a - x}{x}}$$$ med avseende på $$$x$$$

Bestäm $$$\int \sqrt{\frac{a - x}{x}}\, dx$$$.

Inmatningen skrivs om: $$$\int{\sqrt{\frac{a - x}{x}} d x}=\int{\frac{\sqrt{a - x}}{\sqrt{x}} d x}$$$.

Låt $$$u=\sqrt{x}$$$ vara.

Då $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (stegen kan ses »), och vi har att $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

Alltså,

$${\color{red}{\int{\frac{\sqrt{a - x}}{\sqrt{x}} d x}}} = {\color{red}{\int{2 \sqrt{a - u^{2}} d u}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ med $$$c=2$$$ och $$$f{\left(u \right)} = \sqrt{a - u^{2}}$$$:

$${\color{red}{\int{2 \sqrt{a - u^{2}} d u}}} = {\color{red}{\left(2 \int{\sqrt{a - u^{2}} d u}\right)}}$$

Låt $$$u=\sqrt{a} \sin{\left(v \right)}$$$ vara.

Då $$$du=\left(\sqrt{a} \sin{\left(v \right)}\right)^{\prime }dv = \sqrt{a} \cos{\left(v \right)} dv$$$ (stegen kan ses »).

Det följer också att $$$v=\operatorname{asin}{\left(\frac{u}{\sqrt{a}} \right)}$$$.

Alltså,

$$$\sqrt{- u ^{2} + a} = \sqrt{- a \sin^{2}{\left( v \right)} + a}$$$

Använd identiteten $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

$$$\sqrt{- a \sin^{2}{\left( v \right)} + a}=\sqrt{a} \sqrt{1 - \sin^{2}{\left( v \right)}}=\sqrt{a} \sqrt{\cos^{2}{\left( v \right)}}$$$

Om vi antar att $$$\cos{\left( v \right)} \ge 0$$$, erhåller vi följande:

$$$\sqrt{a} \sqrt{\cos^{2}{\left( v \right)}} = \sqrt{a} \cos{\left( v \right)}$$$

Alltså,

$$2 {\color{red}{\int{\sqrt{a - u^{2}} d u}}} = 2 {\color{red}{\int{a \cos^{2}{\left(v \right)} d v}}}$$

Använd potensreduceringsformeln $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ med $$$\alpha= v $$$:

$$2 {\color{red}{\int{a \cos^{2}{\left(v \right)} d v}}} = 2 {\color{red}{\int{\frac{a \left(\cos{\left(2 v \right)} + 1\right)}{2} d v}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(v \right)} = a \left(\cos{\left(2 v \right)} + 1\right)$$$:

$$2 {\color{red}{\int{\frac{a \left(\cos{\left(2 v \right)} + 1\right)}{2} d v}}} = 2 {\color{red}{\left(\frac{\int{a \left(\cos{\left(2 v \right)} + 1\right) d v}}{2}\right)}}$$

Expand the expression:

$${\color{red}{\int{a \left(\cos{\left(2 v \right)} + 1\right) d v}}} = {\color{red}{\int{\left(a \cos{\left(2 v \right)} + a\right)d v}}}$$

Integrera termvis:

$${\color{red}{\int{\left(a \cos{\left(2 v \right)} + a\right)d v}}} = {\color{red}{\left(\int{a d v} + \int{a \cos{\left(2 v \right)} d v}\right)}}$$

Tillämpa konstantregeln $$$\int c\, dv = c v$$$ med $$$c=a$$$:

$$\int{a \cos{\left(2 v \right)} d v} + {\color{red}{\int{a d v}}} = \int{a \cos{\left(2 v \right)} d v} + {\color{red}{a v}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ med $$$c=a$$$ och $$$f{\left(v \right)} = \cos{\left(2 v \right)}$$$:

$$a v + {\color{red}{\int{a \cos{\left(2 v \right)} d v}}} = a v + {\color{red}{a \int{\cos{\left(2 v \right)} d v}}}$$

Låt $$$w=2 v$$$ vara.

Då $$$dw=\left(2 v\right)^{\prime }dv = 2 dv$$$ (stegen kan ses »), och vi har att $$$dv = \frac{dw}{2}$$$.

Alltså,

$$a v + a {\color{red}{\int{\cos{\left(2 v \right)} d v}}} = a v + a {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}$$

Tillämpa konstantfaktorregeln $$$\int c f{\left(w \right)}\, dw = c \int f{\left(w \right)}\, dw$$$ med $$$c=\frac{1}{2}$$$ och $$$f{\left(w \right)} = \cos{\left(w \right)}$$$:

$$a v + a {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}} = a v + a {\color{red}{\left(\frac{\int{\cos{\left(w \right)} d w}}{2}\right)}}$$

Integralen av cosinus är $$$\int{\cos{\left(w \right)} d w} = \sin{\left(w \right)}$$$:

$$a v + \frac{a {\color{red}{\int{\cos{\left(w \right)} d w}}}}{2} = a v + \frac{a {\color{red}{\sin{\left(w \right)}}}}{2}$$

Kom ihåg att $$$w=2 v$$$:

$$a v + \frac{a \sin{\left({\color{red}{w}} \right)}}{2} = a v + \frac{a \sin{\left({\color{red}{\left(2 v\right)}} \right)}}{2}$$

Kom ihåg att $$$v=\operatorname{asin}{\left(\frac{u}{\sqrt{a}} \right)}$$$:

$$\frac{a \sin{\left(2 {\color{red}{v}} \right)}}{2} + a {\color{red}{v}} = \frac{a \sin{\left(2 {\color{red}{\operatorname{asin}{\left(\frac{u}{\sqrt{a}} \right)}}} \right)}}{2} + a {\color{red}{\operatorname{asin}{\left(\frac{u}{\sqrt{a}} \right)}}}$$

Kom ihåg att $$$u=\sqrt{x}$$$:

$$\frac{a \sin{\left(2 \operatorname{asin}{\left(\frac{{\color{red}{u}}}{\sqrt{a}} \right)} \right)}}{2} + a \operatorname{asin}{\left(\frac{{\color{red}{u}}}{\sqrt{a}} \right)} = \frac{a \sin{\left(2 \operatorname{asin}{\left(\frac{{\color{red}{\sqrt{x}}}}{\sqrt{a}} \right)} \right)}}{2} + a \operatorname{asin}{\left(\frac{{\color{red}{\sqrt{x}}}}{\sqrt{a}} \right)}$$

Alltså,

$$\int{\frac{\sqrt{a - x}}{\sqrt{x}} d x} = \frac{a \sin{\left(2 \operatorname{asin}{\left(\frac{\sqrt{x}}{\sqrt{a}} \right)} \right)}}{2} + a \operatorname{asin}{\left(\frac{\sqrt{x}}{\sqrt{a}} \right)}$$

Använd formlerna $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$ för att förenkla uttrycket:

$$\int{\frac{\sqrt{a - x}}{\sqrt{x}} d x} = \sqrt{a} \sqrt{x} \sqrt{1 - \frac{x}{a}} + a \operatorname{asin}{\left(\frac{\sqrt{x}}{\sqrt{a}} \right)}$$

Förenkla ytterligare:

$$\int{\frac{\sqrt{a - x}}{\sqrt{x}} d x} = \sqrt{a} \sqrt{x} \sqrt{\frac{a - x}{a}} + a \operatorname{asin}{\left(\frac{\sqrt{x}}{\sqrt{a}} \right)}$$

Lägg till integrationskonstanten:

$$\int{\frac{\sqrt{a - x}}{\sqrt{x}} d x} = \sqrt{a} \sqrt{x} \sqrt{\frac{a - x}{a}} + a \operatorname{asin}{\left(\frac{\sqrt{x}}{\sqrt{a}} \right)}+C$$

$$$\int \sqrt{\frac{a - x}{x}}\, dx = \left(\sqrt{a} \sqrt{x} \sqrt{\frac{a - x}{a}} + a \operatorname{asin}{\left(\frac{\sqrt{x}}{\sqrt{a}} \right)}\right) + C$$$A